Folded integer encoding

ABSTRACT

Techniques of data compression involve performing a separate compression operation on each set of corresponding bits of a sequence of bit strings in which each bit string represents a number having an upper bound. Advantageously, compressing the sets of corresponding bits produces an improved compression ratio over compressing each number in the sequence. Further, decompression is straightforward as long as sequence order is preserved and the upper bound of each number in the sequence is known.

TECHNICAL FIELD

This description relates to data compression.

BACKGROUND

Many data-acquisition activities result in large sequences of numbersthat are stored in some form of storage media. A common problem involvescompressing the sequence of numbers to save storage space. The solutionto such a problem may involve an entropy encoding scheme such as Huffmanencoding.

SUMMARY

In one general aspect, a computer-implemented method can includereceiving, by processing circuitry of the computer, a first plurality ofbit strings. The computer-implemented method can also include producing,by the processing circuitry, upper bound data based on the firstplurality of bit strings, the upper bound data indicating, for each ofthe first plurality of bit strings, a leftmost bit of that bit string.The computer-implemented method can further include generating, by theprocessing circuitry, a second plurality of bit strings from the firstplurality of bit strings and the upper bound data, each of the secondplurality of bit strings including a bit from a respective bit string ofthe first plurality of bit strings, each bit of each of the secondplurality of bit strings having the same position from the leftmost bitindicated by the upper bound data of its respective bit string of thefirst plurality of bit strings. The computer-implemented method canfurther include performing, by the processing circuitry, a compressionoperation on each of the second plurality of bit strings to (i) producea plurality of compressed bit strings and (ii) reduce an amount ofstorage in a storage device of the computer consumed by the plurality ofcompressed bit strings by the performance of a compression operation oneach of the second plurality of bit strings rather than each of thefirst plurality of bit strings.

The details of one or more implementations are set forth in theaccompanying drawings and the description below. Other features will beapparent from the description and drawings, and from the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram that illustrates an example electronic environmentin which improved techniques described herein may be implemented.

FIG. 2 is a flow chart that illustrates an example method ofimplementing the improved techniques shown in FIG. 1.

FIG. 3A is a diagram that illustrates an example set of bit stringsshown in FIG. 1.

FIG. 3B is a diagram that illustrates an example set of folded bitstrings derived from the bit strings shown in FIG. 3B.

FIG. 4 is a flow chart that illustrates an example process ofcompressing data shown in FIG. 1.

FIG. 5 is a flow chart that illustrates an example process ofdecompressing data shown in FIG. 1.

FIG. 6 illustrates an example of a computer device and a mobile computerdevice that can be used with circuits described here.

DETAILED DESCRIPTION

A large amount of numerical data such as that representing subdivisionsof a point cloud uses an enormous amount of storage and processingresources. Conventional compression techniques such as arithmetic codingare able to reduce the storage and processing requirements somewhat.However, such conventional compression techniques do not take intoaccount certain structural properties of such subdivisions. Therefore,for data such as point cloud subdivision data, conventional compressiontechniques may be suboptimal.

In accordance with the implementations described herein and in contrastwith the above-described conventional techniques of data compression,improved techniques involve performing a separate compression operationon each set of corresponding bits of a sequence of bit strings in whicheach bit string represents a number having an upper bound.Advantageously, compressing the sets of corresponding bits produces animproved compression ratio over compressing each bit string in thesequence. Further, decompression is straightforward as long as sequenceorder is preserved and the upper bound of each number in the sequence isknown.

FIG. 1 is a diagram that illustrates an example electronic environment100 in which the above-described improved techniques may be implemented.As shown, in FIG. 1, the electronic environment 100 includes acompression computer 120.

The compression computer 120 is configured to receive a sequence of bitstrings, perform compression operations to reduce the burden of storingthe bit strings, and perform decompression operations to recover thesequence of bit strings. In some implementations, the compressioncomputer 120 may be a server computer but can also be a desktopcomputer, a laptop computer, a tablet computer, a smartphone, or thelike.

The compression computer 120 includes a network interface 122, one ormore processing units 124, and memory 126. The network interface 122includes, for example, Ethernet adaptors, Token Ring adaptors, and thelike, for converting electronic and/or optical signals received from thenetwork 170 to electronic form for use by the point cloud compressioncomputer 120. The set of processing units 124 include one or moreprocessing chips and/or assemblies. The memory 126 includes bothvolatile memory (e.g., RAM) and non-volatile memory, such as one or moreROMs, disk drives, solid state drives, and the like. The set ofprocessing units 124 and the memory 126 together form control circuitry,which is configured and arranged to carry out various methods andfunctions as described herein.

In some embodiments, one or more of the components of the compressioncomputer 120 can be, or can include processors (e.g., processing units124) configured to process instructions stored in the memory 126.Examples of such instructions as depicted in FIG. 1 include a raw datamanager 130, a bit string folding manager 140, a compression manager150, and a decompression manager 160. Further, as illustrated in FIG. 1,the memory 126 is configured to store various data, which is describedwith respect to the respective managers that use such data.

The raw data manager 130 is configured to receive and store bit strings134. The bit strings 134 may consume a large amount of storage.Accordingly, in some implementations, these data are stored innon-volatile storage media such as a magnetic disk drive, a solid-statedrive, and the like. However, in other implementations, there may not bea need for the raw data manager 130 as the bit strings 134 may alreadybe present in the memory 126 (i.e., generated by the processing units124).

Further, in some implementations, each bit string 134 has acorresponding upper bound which may be a number of bits occupied by someprevious bit string 134. For example, in cases where the bit stringsrepresent numbers of points within subdivisions of a bounding box asdescribed above, such an upper bound may be indicative of a maximumestimate of a number of points within a subdivision. For example, theupper bound may be the number of bits in a bit string representing anumber of points in a previous subdivision. For example, suppose that abounding box enclosing N points is split in two (not necessarily equal)pieces along one direction, e.g., the x-direction. There are N1 pointsto the left of the split and N−N1 points to the right of the split. Forthe N1 points to the left, N represents an upper bound, with the numberof bits and therefore a leftmost bit being the integer part of log₂ N.

In some implementations, the bit strings 134 are received by thecompression computer 120 in a particular order corresponding to asubdivision sequence known in advance. For example, the bounding box maybe split in the x-direction first, then the y-direction, then thez-direction, and then repeating this pattern. However, in otherimplementations, the bounding box may be split in any order. In thiscase, the compression computer 120 might receive additional dataspecifying the subdivision sequence.

Finally, the raw data manager 130 is configured to delete the bitstrings 134 upon folding operations and/or compression operations beingperformed.

The bit string folding manager 140 is configured to generate a set offolded bit strings 142 from the bit strings 134. Each folded bit stringcontains a bit from at least one of the bit strings 134 according to theposition of that bit. In some implementations, the second plurality ofbit strings is generated from the first plurality of bit strings in asequence. In this case, a bit from a particular bit string 134 is addedeach of the folded bit strings 142 according to whether the leftmost bitof the particular bit string (expressed in the upper bound data) isgreater than or less than the sequence number of that folded bit string142 in the sequence.

Along these lines, the first folded bit string 142 would include theleftmost bit from each of the bit strings 134. The second folded bitstring 142 would include the next leftmost bit from each of the bitstrings 134, and so on.

It should be noted that some folded bit strings 142 may not contain bitsof all of the bit strings 134 because the bit strings 134 may havedifferent lengths. Again, such lengths are expressed in the upper bounddata 146. The upper bound data 146 are used by the decompression managerto unfold decompressed bit strings.

It should also be noted that the upper bound data 146 is not typicallystored in the memory 126. Rather, the compression computer 120 is, e.g.,configured to generate an upper bound datum in place by computing thelogarithm base two of a previous bit string 134. However, in someimplementations, the upper bound data is stored in the memory 126. Infurther implementations, the upper bound data is stored in memoryexternal to the compression computer 120.

The compression manager 150 is configured to apply an encoding scheme tocompress the folded bit strings 142 into compressed bit strings 154. Asdepicted in FIG. 1, the compression manager 150 uses arithmetic decoders152 to perform the compression on each of the folded bit strings 142. Insome implementations, the arithmetic decoders 152 are adaptive, i.e.,the probability distributions of the symbols used by the encoders at apresent time depends on the frequencies of the symbols at previoustimes. However, in other implementations, the compression manager mayuse other encoders, i.e., Huffman encoders, asymmetrical number systems,and the like.

The decompression manager 160 is configured to apply a decoding schemeto decompress the compressed bit strings 154 to produce decompressed bitstrings 164. If the decompression scheme functions properly, then thedecompressed bit strings 164 are essentially equivalent to the foldedbit strings 142. The decoders, in this case, arithmetic decoders 162,correspond respectively to the arithmetic encoders 152.

The decompression manager 160 is also configured to perform an unfoldingoperation on the decompressed bit strings to recover the original bitstrings 134 to produce unfolded bit strings 166. If the unfoldingfunctions properly, then the unfolded bit strings 166 are equivalent tothe bit strings 134. In performing the unfolding operation, thedecompression manager 160 is configured to use the upper bound data 146to determine where each unfolded bit string 166 terminates.

In some implementations, the memory 126 can be any type of memory suchas a random-access memory, a disk drive memory, flash memory, and/or soforth. In some implementations, the memory 126 can be implemented asmore than one memory component (e.g., more than one RAM component ordisk drive memory) associated with the components of the user device120. In some implementations, the memory 126 can be a database memory.In some implementations, the memory 126 can be, or can include, anon-local memory. For example, the memory 126 can be, or can include, amemory shared by multiple devices (not shown). In some implementations,the memory 126 can be associated with a server device (not shown) withina network and configured to serve the components of the user device 120.

The components (e.g., modules, processing units 124) of the compressioncomputer 120 can be configured to operate based on one or more platforms(e.g., one or more similar or different platforms) that can include oneor more types of hardware, software, firmware, operating systems,runtime libraries, and/or so forth. In some implementations, thecomponents of the compression computer 120 can be configured to operatewithin a cluster of devices (e.g., a server farm). In such animplementation, the functionality and processing of the components ofthe compression computer 120 can be distributed to several devices ofthe cluster of devices.

The components of the compression computer 120 can be, or can include,any type of hardware and/or software configured to process attributes.In some implementations, one or more portions of the components shown inthe components of the compression computer 120 in FIG. 1 can be, or caninclude, a hardware-based module (e.g., a digital signal processor(DSP), a field programmable gate array (FPGA), a memory), a firmwaremodule, and/or a software-based module (e.g., a module of computer code,a set of computer-readable instructions that can be executed at acomputer). For example, in some implementations, one or more portions ofthe components of the compression computer 120 can be, or can include, asoftware module configured for execution by at least one processor (notshown). In some implementations, the functionality of the components canbe included in different modules and/or different components than thoseshown in FIG. 1.

Although not shown, in some implementations, the components of thecompression computer 120 (or portions thereof) can be configured tooperate within, for example, a data center (e.g., a cloud computingenvironment), a computer system, one or more server/host devices, and/orso forth. In some implementations, the components of the compressioncomputer 120 (or portions thereof) can be configured to operate within anetwork. Thus, the components of the compression computer 120 (orportions thereof) can be configured to function within various types ofnetwork environments that can include one or more devices and/or one ormore server devices. For example, the network can be, or can include, alocal area network (LAN), a wide area network (WAN), and/or so forth.The network can be, or can include, a wireless network and/or wirelessnetwork implemented using, for example, gateway devices, bridges,switches, and/or so forth. The network can include one or more segmentsand/or can have portions based on various protocols such as InternetProtocol (IP) and/or a proprietary protocol. The network can include atleast a portion of the Internet.

In some embodiments, one or more of the components of the compressioncomputer 120 can be, or can include, processors configured to processinstructions stored in a memory. For example, the raw data manager 130(and/or a portion thereof), the bit string folding manager 140 (and/or aportion thereof), the compression manager 150 (and/or a portionthereof), and the decompression manager 160 (and/or a portion thereof)can be a combination of a processor and a memory configured to executeinstructions related to a process to implement one or more functions.

FIG. 2 is a flow chart that illustrates an example method 200 ofimplementing the improved techniques shown in FIG. 1. The method 200 maybe performed by software constructs described in connection with FIG. 1,which reside in memory 126 of the compression computer 120 and are runby the set of processing units 124.

At 202, a first plurality of bit strings (e.g., bit strings 134) isreceived. In some implementations, each bit string of the firstplurality of bit strings may represent a number of points within asubdivision of a bounding box enclosing a point cloud.

At 204, upper bound data is produced from the received first pluralityof strings. The upper bound data indicates a leftmost bit of each of thefirst plurality of bit strings that may not be zero, i.e., all bits tothe left of the leftmost bit are guaranteed to be zero.

At 206, a second plurality of bit strings (e.g., folded bit strings 142)is generated from the first plurality of bit strings and the upper bounddata. Each of the second plurality of bit strings includes a bit from arespective bit string of the first plurality of bit strings. Each bit ofeach of the second plurality of bit strings having the same positionfrom the leftmost bit indicated by the upper bound data of itsrespective bit string of the first plurality of bit strings.

At 208, a compression operation is performed on each of the secondplurality of bit strings to (i) produce a plurality of compressed bitstrings and (ii) reduce an amount of storage in a storage device of acompression computer 120 consumed by the plurality of compressed bitstrings by the performance of the compression operation on each of thesecond plurality of bit strings rather than each of the first pluralityof bit strings.

FIG. 3A is a diagram that illustrates an example set of bit strings 134.FIG. 3A depicts M numbers 310(1), 310(2), . . . , 310(M), represented asbit strings. Each number, e.g., number 310(1), has a number of bits asgiven in the upper bound data 320 (5 in this case). The leftmost bit isshown to have a box around that bit. The leftmost bit of number 310(1)in the upper bound data 320 may be determined by a location of thenumber 310(1) within the bounding box 190 as described above, e.g., bythe subdivision of the bounding box 190 in which the number 310(1) islocated.

FIG. 3C is a diagram that illustrates an example set of folded bitstrings derived from the bit strings in FIG. 3B. Here, each folded bitstring such as folded bit string 330(1) includes a bit from each bitstring 310(1), . . . , 310(M). In the case of folded bit string 310(1),each bit is the leftmost bit from each bit string 310(1), . . . ,310(M). Other folded bit strings may have fewer bits according to howmany bits are in each bit string 310(1), . . . , 310(M). For example,folded bit string 330(3) has one bit less than folded bit strings 330(1and 330(2). Subsequent folded bit strings 330(4), . . . , 330(N) mayhave fewer bits still.

It is these folded bit strings 330(1), . . . , 330(N) that arecompressed using, e.g., arithmetic encoders 340(1), . . . , 340(N) toproduce compressed bit strings 350(1), . . . , 350(N). Each differentarithmetic encoder, e.g., 340(1) provides a different context. Thenumber of different contexts is determined here by the length of thebounding box 190. For example, the arithmetic encoder 340(1) may have afirst set of symbols and a probability distribution of those symbols,while the arithmetic encoder 340(2) has a second set of symbols and aprobability distribution of those symbols, and so on.

An advantage of performing compression in this way is that compressingthe folded bit strings 330(1), . . . , 330(N) may result in a highercompression ratio than compressing bit strings 310(1), . . . , 310(M). Areason for this can be attributed to the fact the upper bound dataalready helps avoid encoding bits that are known to be zero. However, inmany cases the upper bound is not sharp; this implies that the leadingbits of the encoded part of a bit string are more likely to be zero.That is, it is advantageous to aggregate these bits in one context, i.e.bit string, and encode it separate from the others. This reason appliessimilarly for second bits and so forth.

FIG. 4 is a flow chart that illustrates an example process 400 ofdecompression to recover original bit strings. The process 400 may beperformed by software constructs described in connection with FIG. 1,which reside in memory 126 of the user device 120 and are run by the setof processing units 124.

At 402, the compression computer 120 receives a large number of bitstrings. The bit strings may originate from a source external to thecompression computer 120. The bit strings may represent numbers derivedfrom some physical process, e.g., distributions of points of a pointcloud within a bounding box.

At 404, the compression computer 120 produces upper bound data from thebit strings it has received. For example, in the case where the bitstrings represents distributions of points of a point cloud withinvarious subregions of a bounding box, the first bit string mayrepresents the total number of points. This total number may be an upperbound of the number represented by the second bit string, which numberis the number of points in a first subdivision of the bounding box(e.g., along the x-axis). In turn, the number represented by the secondbit string may be an upper bound of the third bit string, and so on. Theupper bound data may be represented in terms of a maximum number of bitsthat a bit string may have.

At 406, the compression computer 120 generates folded bit strings fromthe bit strings and the upper bound data, as illustrated in FIGS. 3A and3B.

At 408, the compression computer 120 encodes the folded bit strings toform compressed, folded bit strings.

FIG. 5 is a flow chart that illustrates an example process 500 ofdecompression to recover original bit strings. The process 500 may beperformed by software constructs described in connection with FIG. 1,which reside in memory 126 of the user device 120 and are run by the setof processing units 124.

At 502, the compression computer 120 receives N compressed bit strings.These N compressed bit strings are the result of performing acompression operation separately on each of a set of folded bit strings,e.g., folded bit strings 330(1), . . . , 330(N). In the examplesdiscussed herein, the compression operations each involve application ofan arithmetic encoding scheme.

At 504, the decompression manager 160 performs a decoding operation oneach of the received compressed bit strings to produce N folded bitstrings. In this case, for each compressed bit string, the decompressionmanager 160 uses an arithmetic decoder corresponding to the arithmeticencoder that was applied to a folded bit string to produce thatcompressed bit string. In other implementations, there may be N encodersand corresponding decoders that may be applied to the bit strings inparallel.

At 506, the compression computer 120 produces upper bound data used tounfold the folded bit strings, e.g., computes the logarithm base two ofthe number represented by a previously decoded bit string. In someimplementations, the previously decoded bit string is immediately prior,while in other implementations the previously decoded bit string mayrepresent a combination of other bit strings depending on the context inwhich this module is placed.

At 508, the decompression manager 160 unfolds the folded bit stringsusing the upper bound data. For example, consider the folded bit strings330(1), . . . , 330(N) and the upper bound data 320. As the first upperbound datum indicates 5 bits, then the decompression manager 160 selectsthe first bit from each of the first 5 folded bit strings 330(1), . . ., 330(5). The decompression manager 160 may then (e.g., remove) deletethese bits from those folded bit strings. The next upper bound datumindicates 3 bits; in this case, the decompression manager 160 selectsthe first bit from each of the first 3 folded bit strings 330(1), . . ., 330(3). This process is repeated until all of the bits of the foldedbit strings have been selected.

The above examples assumed that the bit strings received by thecompression computer 120 resulted from counting points in successivesubdivisions of a bounding box enclosing a point cloud. However, thetechniques discussed herein apply to any sequences of integers in whichthe numbers are smaller than an upper bound and in which smaller numbersare much more likely than larger numbers.

FIG. 6 illustrates an example of a generic computer device 600 and ageneric mobile computer device 650, which may be used with thetechniques described here.

As shown in FIG. 6, computing device 600 is intended to representvarious forms of digital computers, such as laptops, desktops,workstations, personal digital assistants, servers, blade servers,mainframes, and other appropriate computers. Computing device 650 isintended to represent various forms of mobile devices, such as personaldigital assistants, cellular telephones, smart phones, and other similarcomputing devices. The components shown here, their connections andrelationships, and their functions, are meant to be exemplary only, andare not meant to limit implementations of the inventions describedand/or claimed in this document.

Computing device 600 includes a processor 602, memory 604, a storagedevice 606, a high-speed interface 608 connecting to memory 604 andhigh-speed expansion ports 610, and a low speed interface 612 connectingto low speed bus 614 and storage device 606. Each of the components 602,604, 606, 608, 610, and 612, are interconnected using various busses,and may be mounted on a common motherboard or in other manners asappropriate. The processor 602 can process instructions for executionwithin the computing device 600, including instructions stored in thememory 604 or on the storage device 606 to display graphical informationfor a GUI on an external input/output device, such as display 616coupled to high speed interface 608. In other implementations, multipleprocessors and/or multiple buses may be used, as appropriate, along withmultiple memories and types of memory. Also, multiple computing devices600 may be connected, with each device providing portions of thenecessary operations (e.g., as a server bank, a group of blade servers,or a multi-processor system).

The memory 604 stores information within the computing device 600. Inone implementation, the memory 604 is a volatile memory unit or units.In another implementation, the memory 604 is a non-volatile memory unitor units. The memory 604 may also be another form of computer-readablemedium, such as a magnetic or optical disk.

The storage device 606 is capable of providing mass storage for thecomputing device 600. In one implementation, the storage device 606 maybe or contain a computer-readable medium, such as a floppy disk device,a hard disk device, an optical disk device, or a tape device, a flashmemory or other similar solid state memory device, or an array ofdevices, including devices in a storage area network or otherconfigurations. A computer program product can be tangibly embodied inan information carrier. The computer program product may also containinstructions that, when executed, perform one or more methods, such asthose described above. The information carrier is a computer- ormachine-readable medium, such as the memory 604, the storage device 606,or memory on processor 602.

The high speed controller 608 manages bandwidth-intensive operations forthe computing device 500, while the low speed controller 612 manageslower bandwidth-intensive operations. Such allocation of functions isexemplary only. In one implementation, the high-speed controller 608 iscoupled to memory 604, display 616 (e.g., through a graphics processoror accelerator), and to high-speed expansion ports 610, which may acceptvarious expansion cards (not shown). In the implementation, low-speedcontroller 612 is coupled to storage device 506 and low-speed expansionport 614. The low-speed expansion port, which may include variouscommunication ports (e.g., USB, Bluetooth, Ethernet, wireless Ethernet)may be coupled to one or more input/output devices, such as a keyboard,a pointing device, a scanner, or a networking device such as a switch orrouter, e.g., through a network adapter.

The computing device 600 may be implemented in a number of differentforms, as shown in the figure. For example, it may be implemented as astandard server 620, or multiple times in a group of such servers. Itmay also be implemented as part of a rack server system 624. Inaddition, it may be implemented in a personal computer such as a laptopcomputer 622. Alternatively, components from computing device 600 may becombined with other components in a mobile device (not shown), such asdevice 650. Each of such devices may contain one or more of computingdevice 600, 650, and an entire system may be made up of multiplecomputing devices 600, 650 communicating with each other.

Computing device 650 includes a processor 652, memory 664, aninput/output device such as a display 654, a communication interface666, and a transceiver 668, among other components. The device 650 mayalso be provided with a storage device, such as a microdrive or otherdevice, to provide additional storage. Each of the components 650, 652,664, 654, 666, and 668, are interconnected using various buses, andseveral of the components may be mounted on a common motherboard or inother manners as appropriate.

The processor 652 can execute instructions within the computing device650, including instructions stored in the memory 664. The processor maybe implemented as a chipset of chips that include separate and multipleanalog and digital processors. The processor may provide, for example,for coordination of the other components of the device 650, such ascontrol of user interfaces, applications run by device 650, and wirelesscommunication by device 650.

Processor 652 may communicate with a user through control interface 658and display interface 656 coupled to a display 654. The display 654 maybe, for example, a TFT LCD (Thin-Film-Transistor Liquid Crystal Display)or an OLED (Organic Light Emitting Diode) display, or other appropriatedisplay technology. The display interface 656 may comprise appropriatecircuitry for driving the display 654 to present graphical and otherinformation to a user. The control interface 658 may receive commandsfrom a user and convert them for submission to the processor 652. Inaddition, an external interface 662 may be provided in communicationwith processor 652, so as to enable near area communication of device650 with other devices. External interface 662 may provide, for example,for wired communication in some implementations, or for wirelesscommunication in other implementations, and multiple interfaces may alsobe used.

The memory 664 stores information within the computing device 650. Thememory 664 can be implemented as one or more of a computer-readablemedium or media, a volatile memory unit or units, or a non-volatilememory unit or units. Expansion memory 674 may also be provided andconnected to device 650 through expansion interface 672, which mayinclude, for example, a SIMM (Single In Line Memory Module) cardinterface. Such expansion memory 674 may provide extra storage space fordevice 650, or may also store applications or other information fordevice 650. Specifically, expansion memory 674 may include instructionsto carry out or supplement the processes described above, and mayinclude secure information also. Thus, for example, expansion memory 674may be provided as a security module for device 650, and may beprogrammed with instructions that permit secure use of device 650. Inaddition, secure applications may be provided via the SIMM cards, alongwith additional information, such as placing identifying information onthe SIMM card in a non-hackable manner.

The memory may include, for example, flash memory and/or NVRAM memory,as discussed below. In one implementation, a computer program product istangibly embodied in an information carrier. The computer programproduct contains instructions that, when executed, perform one or moremethods, such as those described above. The information carrier is acomputer- or machine-readable medium, such as the memory 664, expansionmemory 674, or memory on processor 652, that may be received, forexample, over transceiver 668 or external interface 662.

Device 650 may communicate wirelessly through communication interface666, which may include digital signal processing circuitry wherenecessary. Communication interface 666 may provide for communicationsunder various modes or protocols, such as GSM voice calls, SMS, EMS, orMMS messaging, CDMA, TDMA, PDC, WCDMA, CDMA2000, or GPRS, among others.Such communication may occur, for example, through radio-frequencytransceiver 668. In addition, short-range communication may occur, suchas using a Bluetooth, WiFi, or other such transceiver (not shown). Inaddition, GPS (Global Positioning System) receiver module 670 mayprovide additional navigation- and location-related wireless data todevice 650, which may be used as appropriate by applications running ondevice 650.

Device 650 may also communicate audibly using audio codec 660, which mayreceive spoken information from a user and convert it to usable digitalinformation. Audio codec 660 may likewise generate audible sound for auser, such as through a speaker, e.g., in a handset of device 650. Suchsound may include sound from voice telephone calls, may include recordedsound (e.g., voice messages, music files, etc.) and may also includesound generated by applications operating on device 650.

The computing device 650 may be implemented in a number of differentforms, as shown in the figure. For example, it may be implemented as acellular telephone 680. It may also be implemented as part of a smartphone 682, personal digital assistant, or other similar mobile device.

Various implementations of the systems and techniques described here canbe realized in digital electronic circuitry, integrated circuitry,specially designed ASICs (application specific integrated circuits),computer hardware, firmware, software, and/or combinations thereof.These various implementations can include implementation in one or morecomputer programs that are executable and/or interpretable on aprogrammable system including at least one programmable processor, whichmay be special or general purpose, coupled to receive data andinstructions from, and to transmit data and instructions to, a storagesystem, at least one input device, and at least one output device.

These computer programs (also known as programs, software, softwareapplications or code) include machine instructions for a programmableprocessor, and can be implemented in a high-level procedural and/orobject-oriented programming language, and/or in assembly/machinelanguage. As used herein, the terms “machine-readable medium”“computer-readable medium” refers to any computer program product,apparatus and/or device (e.g., magnetic discs, optical disks, memory,Programmable Logic Devices (PLDs)) used to provide machine instructionsand/or data to a programmable processor, including a machine-readablemedium that receives machine instructions as a machine-readable signal.The term “machine-readable signal” refers to any signal used to providemachine instructions and/or data to a programmable processor.

To provide for interaction with a user, the systems and techniquesdescribed here can be implemented on a computer having a display device(e.g., a CRT (cathode ray tube) or LCD (liquid crystal display) monitor)for displaying information to the user and a keyboard and a pointingdevice (e.g., a mouse or a trackball) by which the user can provideinput to the computer. Other kinds of devices can be used to provide forinteraction with a user as well; for example, feedback provided to theuser can be any form of sensory feedback (e.g., visual feedback,auditory feedback, or tactile feedback); and input from the user can bereceived in any form, including acoustic, speech, or tactile input.

The systems and techniques described here can be implemented in acomputing system that includes a back end component (e.g., as a dataserver), or that includes a middleware component (e.g., an applicationserver), or that includes a front end component (e.g., a client computerhaving a graphical user interface or a Web browser through which a usercan interact with an implementation of the systems and techniquesdescribed here), or any combination of such back end, middleware, orfront end components. The components of the system can be interconnectedby any form or medium of digital data communication (e.g., acommunication network). Examples of communication networks include alocal area network (“LAN”), a wide area network (“WAN”), and theInternet.

The computing system can include clients and servers. A client andserver are generally remote from each other and typically interactthrough a communication network. The relationship of client and serverarises by virtue of computer programs running on the respectivecomputers and having a client-server relationship to each other.

A number of embodiments have been described. Nevertheless, it will beunderstood that various modifications may be made without departing fromthe spirit and scope of the specification.

It will also be understood that when an element is referred to as beingon, connected to, electrically connected to, coupled to, or electricallycoupled to another element, it may be directly on, connected or coupledto the other element, or one or more intervening elements may bepresent. In contrast, when an element is referred to as being directlyon, directly connected to or directly coupled to another element, thereare no intervening elements present. Although the terms directly on,directly connected to, or directly coupled to may not be used throughoutthe detailed description, elements that are shown as being directly on,directly connected or directly coupled can be referred to as such. Theclaims of the application may be amended to recite exemplaryrelationships described in the specification or shown in the figures.

While certain features of the described implementations have beenillustrated as described herein, many modifications, substitutions,changes and equivalents will now occur to those skilled in the art. Itis, therefore, to be understood that the appended claims are intended tocover all such modifications and changes as fall within the scope of theimplementations. It should be understood that they have been presentedby way of example only, not limitation, and various changes in form anddetails may be made. Any portion of the apparatus and/or methodsdescribed herein may be combined in any combination, except mutuallyexclusive combinations. The implementations described herein can includevarious combinations and/or sub-combinations of the functions,components and/or features of the different implementations described.

In addition, the logic flows depicted in the figures do not require theparticular order shown, or sequential order, to achieve desirableresults. In addition, other steps may be provided, or steps may beeliminated, from the described flows, and other components may be addedto, or removed from, the described systems. Accordingly, otherembodiments are within the scope of the following claims.

What is claimed is:
 1. A computer-implemented method, comprising:receiving, by processing circuitry of the computer, a first plurality ofbit strings; producing, by the processing circuitry, upper bound databased on the first plurality of bit strings, the upper bound dataindicating, for each of the first plurality of bit strings, a leftmostbit of that bit string; generating, by the processing circuitry, asecond plurality of bit strings from the first plurality of bit stringsand the upper bound data, each of the second plurality of bit stringsincluding a bit from a respective bit string of the first plurality ofbit strings, each bit of each of the second plurality of bit stringshaving the same position from the leftmost bit indicated by the upperbound data of its respective bit string of the first plurality of bitstrings; and performing, by the processing circuitry, a compressionoperation on each of the second plurality of bit strings to (i) producea plurality of compressed bit strings and (ii) reduce an amount ofstorage in a storage device of the computer consumed by the plurality ofcompressed bit strings by the performance of a compression operation oneach of the second plurality of bit strings rather than each of thefirst plurality of bit strings.
 2. The computer-implemented method as inclaim 1, wherein each of the first plurality of bit strings represents anumber of points of a point cloud enclosed within a respectivesubdivision of a series of subdivisions of a bounding box, and whereinproducing the upper bound data includes, for each of the first pluralityof bit strings, generating a number of bits in a bit string representinga number of points in a previous subdivision of the bounding box.
 3. Thecomputer-implemented method as in claim 2, wherein generating the secondplurality of bit strings from the first plurality of bit stringsincludes: defining a sequence in which the second plurality of bitstrings is generated from the first plurality of bit strings, each ofthe second plurality of bit strings having a sequence number in thesequence; and for each of the second plurality of bit strings, adding abit from a particular bit string of the first plurality of bit stringsaccording to whether the leftmost bit of the particular bit string isgreater than or less than the sequence number of that bit string of thesecond plurality of bit strings in the sequence.
 4. Thecomputer-implemented method as in claim 2, further comprising receivingsequencing data indicating a sequential order into which the series ofsubdivisions of the bounding box were generated.
 5. Thecomputer-implemented method as in claim 2, further comprising:performing a decompression operation on each of the compressed bitstrings to produce the second plurality of bit strings; and performing abit string unfolding operation on the second plurality of bit stringsusing the upper bound data to produce the first plurality of bitstrings.
 6. The computer-implemented method as in claim 1, whereinperforming the compression operation includes applying a respectivearithmetic encoder to each of the second plurality of bit strings toproduce, for each of the plurality of compressed bit strings, a singlenumber as a compressed bit string.
 7. The computer-implemented method asin claim 1, wherein producing the upper bound data based on the firstplurality of bit strings includes retrieving the upper bound data frommemory of the computer.
 8. A computer program product comprising anontransitive storage medium, the computer program product includingcode that, when executed by processing circuitry, causes the processingcircuitry to perform a method, the method comprising: receiving a firstplurality of bit strings; producing upper bound data based on the firstplurality of bit strings, the upper bound data indicating, for each ofthe first plurality of bit strings, a leftmost bit of that bit string;generating a second plurality of bit strings from the first plurality ofbit strings and the upper bound data, each of the second plurality ofbit strings including a bit from a respective bit string of the firstplurality of bit strings, each bit of each of the second plurality ofbit strings having the same position from the leftmost bit indicated bythe upper bound data of its respective bit string of the first pluralityof bit strings; and performing a compression operation on each of thesecond plurality of bit strings to (i) produce a plurality of compressedbit strings and (ii) reduce an amount of storage in a storage device ofthe computer consumed by the plurality of compressed bit strings by theperformance of a compression operation on each of the second pluralityof bit strings rather than each of the first plurality of bit strings.9. The computer program product as in claim 8, wherein each of the firstplurality of bit strings represents a number of points of a point cloudenclosed within a respective subdivision of a series of subdivisions ofa bounding box, and wherein producing the upper bound data includes, foreach of the first plurality of bit strings, generating a number of bitsin a bit string representing a number of points in a previoussubdivision of the bounding box.
 10. The computer program product as inclaim 9, wherein generating the second plurality of bit strings from thefirst plurality of bit strings includes: defining a sequence in whichthe second plurality of bit strings is generated from the firstplurality of bit strings, each of the second plurality of bit stringshaving a sequence number in the sequence; and for each of the secondplurality of bit strings, adding a bit from a particular bit string ofthe first plurality of bit strings according to whether the leftmost bitof the particular bit string is greater than or less than the sequencenumber of that bit string of the second plurality of bit strings in thesequence.
 11. The computer program product as in claim 9, wherein themethod further comprises receiving sequencing data indicating asequential order into which the series of subdivisions of the boundingbox were generated.
 12. The computer program product as in claim 9,wherein the method further comprises: performing a decompressionoperation on each of the compressed bit strings to produce the secondplurality of bit strings; and performing a bit string unfoldingoperation on the second plurality of bit strings using the upper bounddata to produce the first plurality of bit strings.
 13. The computerprogram product as in claim 8, wherein performing the compressionoperation includes applying a respective arithmetic encoder to each ofthe second plurality of bit strings to produce, for each of theplurality of compressed bit strings, a single number as a compressed bitstring.
 14. The computer program product as in claim 8, whereinproducing the upper bound data based on the first plurality of bitstrings includes retrieving the upper bound data from memory of thecomputer.
 15. An apparatus, comprising: a network interface; memory; andcontrolling circuitry coupled to the memory, the controlling circuitrybeing configured to: receive a first plurality of bit strings; produceupper bound data based on the first plurality of bit strings, the upperbound data indicating, for each of the first plurality of bit strings, aleftmost bit of each of that bit string; generate a second plurality ofbit strings from the first plurality of bit strings and the upper bounddata, each of the second plurality of bit strings including a bit from arespective bit string of the first plurality of bit strings, each bit ofeach of the second plurality of bit strings having the same positionfrom the leftmost bit indicated by the upper bound data of itsrespective bit string of the first plurality of bit strings; and performa compression operation on each of the second plurality of bit stringsto (i) produce a plurality of compressed bit strings and (ii) reduce anamount of storage in a storage device of the computer consumed by theplurality of compressed bit strings by the performance of a compressionoperation on each of the second plurality of bit strings rather thaneach of the first plurality of bit strings.
 16. The apparatus as inclaim 15, wherein each of the first plurality of bit strings representsa number of points of a point cloud enclosed within a respectivesubdivision of a series of subdivisions of a bounding box, and whereinthe controlling circuitry configured to produce the upper bound data isfurther configured to, for each of the first plurality of bit strings,generate a number of bits in a bit string representing a number ofpoints in a previous subdivision of the bounding box.
 17. The apparatusas in claim 16, wherein the controlling circuitry configured to generatethe second plurality of bit strings from the first plurality of bitstrings is further configured to: define a sequence in which the secondplurality of bit strings is generated from the first plurality of bitstrings, each of the second plurality of bit strings having a sequencenumber in the sequence; and for each of the second plurality of bitstrings, add a bit from a particular bit string of the first pluralityof bit strings according to whether the leftmost bit of the particularbit string is greater than or less than the sequence number of that bitstring of the second plurality of bit strings in the sequence.
 18. Theapparatus as in claim 16, wherein the controlling circuitry is furtherconfigured to receive sequencing data indicating a sequential order intowhich the series of subdivisions of the bounding box were generated. 19.The apparatus as in claim 16, wherein the controlling circuitry isfurther configured to: perform a decompression operation on each of thecompressed bit strings to produce the second plurality of bit strings;and perform a bit string unfolding operation on the second plurality ofbit strings using the upper bound data to produce the first plurality ofbit strings.
 20. The apparatus as in claim 15, wherein the controllingcircuitry configured to perform the compression operation is furtherconfigured to apply a respective arithmetic encoder to each of thesecond plurality of bit strings to produce, for each of the plurality ofcompressed bit strings, a single number as a compressed bit string.